Question 1 What is the conjugate of 6 𝑖 6i? Answer 1 If 𝑧 = 𝑎 + 𝑦 𝑖 z=a+yi, then the conjugate of 𝑧 z is 𝑎 − 𝑦 𝑖 a−yi. In this case, 𝑧 = 6 𝑖 z=6i, so the conjugate of 𝑧 z is − 6 𝑖 −6i. Question 2 How do you simplify 8 + 10 𝑖 8 + 6 𝑖 8+6i 8+10i and write it in 𝑎 + 𝑏 𝑖 a+bi form? Answer 1 Multiply the numerator and denominator by the conjugate of the denominator: 8 + 10 𝑖 8 + 6 𝑖 ⋅ 8 − 6 𝑖 8 − 6 𝑖 = ( 8 + 10 𝑖 ) ( 8 − 6 𝑖 ) ( 8 + 6 𝑖 ) ( 8 − 6 𝑖 ) 8+6i 8+10i ⋅ 8−6i 8−6i = (8+6i)(8−6i) (8+10i)(8−6i) Expanding the numerator and denominator: = 64 − 48 𝑖 + 80 𝑖 − 60 𝑖 2 64 − 36 𝑖 2 = 64 + 32 𝑖 + 60 64 + 36 = 124 + 32 𝑖 100 = 64−36i 2 64−48i+80i−60i 2 = 64+36 64+32i+60 = 100 124+32i Simplifying: = 124 100 + 32 𝑖 100 = 1.24 + 0.32 𝑖 = 100 124 + 100 32i =1.24+0.32i So, the result is 1.24 + 0.32 𝑖 1.24+0.32i. Question 3 How do you divide ( 5 − 2 𝑖 ) ÷ ( 3 − 2 𝑖 ) (5−2i)÷(3−2i)? Answer 1 Multiply the numerator and the denominator by the conjugate of the denominator: 5 − 2 𝑖 3 − 2 𝑖 ⋅ 3 + 2 𝑖 3 + 2 𝑖 = ( 5 − 2 𝑖 ) ( 3 + 2 𝑖 ) ( 3 − 2 𝑖 ) ( 3 + 2 𝑖 ) 3−2i 5−2i ⋅ 3+2i 3+2i = (3−2i)(3+2i) (5−2i)(3+2i) Expanding both the numerator and denominator: Numerator: ( 5 − 2 𝑖 ) ( 3 + 2 𝑖 ) = 15 + 10 𝑖 − 6 𝑖 − 4 𝑖 2 = 15 + 4 𝑖 + 4 = 19 + 4 𝑖 (5−2i)(3+2i)=15+10i−6i−4i 2 =15+4i+4=19+4i Denominator: ( 3 − 2 𝑖 ) ( 3 + 2 𝑖 ) = 9 − 4 𝑖 2 = 9 + 4 = 13 (3−2i)(3+2i)=9−4i 2 =9+4=13 Result: 19 + 4 𝑖 13 = 19 13 + 4 13 𝑖 13 19+4i = 13 19 + 13 4 i So, the answer is 19 13 + 4 13 𝑖 13 19 + 13 4 i. Question 4 How do you simplify ( 2 + 4 𝑖 ) ( 2 − 4 𝑖 ) (2+4i)(2−4i)? Answer 1 These two terms are complex conjugates, so their product will be a real number. ( 2 + 4 𝑖 ) ( 2 − 4 𝑖 ) = 4 − 16 𝑖 2 (2+4i)(2−4i)=4−16i 2 Since 𝑖 2 = − 1 i 2 =−1, = 4 + 16 = 20 =4+16=20 The answer is 20 20. Question 5 How do you simplify ( 7 − 3 ) ( 2 − 27 ) (7 −3 )(2 −27 )? Answer 1 Rewrite the square roots of negative numbers using 𝑖 i: ( 7 − 3 ) ( 2 − 27 ) = ( 7 3 ⋅ 𝑖 ) ( 2 ⋅ 3 3 ⋅ 𝑖 ) = ( 7 ⋅ 2 ⋅ 3 ) ( 3 ⋅ 3 ) ( 𝑖 ⋅ 𝑖 ) (7 −3 )(2 −27 )=(7 3 ⋅i)(2⋅3 3 ⋅i)=(7⋅2⋅3)( 3 ⋅ 3 )(i⋅i) Simplify each part: = 42 ⋅ 3 ⋅ ( − 1 ) = 126 ⋅ − 1 = − 126 =42⋅3⋅(−1)=126⋅−1=−126 The answer is − 126 −126.